Found problems: 3882
2022 Indonesia TST, G
Let $AB$ be the diameter of circle $\Gamma$ centred at $O$. Point $C$ lies on ray $\overrightarrow{AB}$. The line through $C$ cuts circle $\Gamma$ at $D$ and $E$, with point $D$ being closer to $C$ than $E$ is. $OF$ is the diameter of the circumcircle of triangle $BOD$. Next, construct $CF$, cutting the circumcircle of triangle $BOD$ at $G$. Prove that $O,A,E,G$ are concyclic.
(Possibly proposed by Pak Wono)
2010 Germany Team Selection Test, 3
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
2017 Iran Team Selection Test, 1
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.
Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.
[i]Proposed by Kasra Ahmadi[/i]
2018 Cono Sur Olympiad, 5
Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.
2010 Kazakhstan National Olympiad, 2
On sides of convex quadrilateral $ABCD$ on external side constructed equilateral triangles $ABK, BCL, CDM, DAN$. Let $P,Q$- midpoints of $BL, AN$ respectively and $X$- circumcenter of $CMD$.
Prove, that $PQ$ perpendicular to $KX$
2018 Balkan MO Shortlist, G6
In a triangle $ABC$ with $AB=AC$, $\omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $BA$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $OAD$ intersects the line $AC$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $DAEH$ is a parallelogram. Line $EH$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $GB$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $GJ$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle.
2016 Croatia Team Selection Test, Problem 3
Let $P$ be a point inside a triangle $ABC$ such that
$$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$
Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.
1999 India Regional Mathematical Olympiad, 3
Let $ABCD$ be a square and $M,N$ points on sides $AB, BC$ respectively such that $\angle MDN = 45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP =RQ$ where $P,Q$ are points of intersection of $AC$ with the lines $MD, ND$.
2013 ELMO Shortlist, 10
Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$.
[i]Proposed by David Stoner[/i]
2015 All-Russian Olympiad, 7
In an acute-angled and not isosceles triangle $ABC,$ we draw the median $AM$ and the height $AH.$
Points $Q$ and $P$ are marked on the lines $AB$ and $AC$, respectively, so that the $QM \perp AC$ and $PM \perp AB$.
The circumcircle of $PMQ$ intersects the line $BC$ for second time at point $X.$ Prove that $BH = CX.$
M. Didin
2007 China Girls Math Olympiad, 2
Let $ ABC$ be an acute triangle. Points $ D$, $ E$, and $ F$ lie on segments $ BC$, $ CA$, and $ AB$, respectively, and each of the three segments $ AD$, $ BE$, and $ CF$ contains the circumcenter of $ ABC$. Prove that if any two of the ratios $ \frac{BD}{DC}$, $ \frac{CE}{EA}$, $ \frac{AF}{FB}$, $ \frac{BF}{FA}$, $ \frac{AE}{EC}$, $ \frac{CD}{DB}$ are integers, then triangle $ ABC$ is isosceles.
2016 Turkey Team Selection Test, 8
All angles of the convex $n$-gon $A_1A_2\dots A_n$ are obtuse, where $n\ge5$. For all $1\le i\le n$, $O_i$ is the circumcenter of triangle $A_{i-1}A_iA_{i+1}$ (where $A_0=A_n$ and $A_{n+1}=A_1$). Prove that the closed path $O_1O_2\dots O_n$ doesn't form a convex $n$-gon.
2009 Indonesia TST, 1
Let $ ABC$ be a triangle. A circle $ P$ is internally tangent to the circumcircle of triangle $ ABC$ at $ A$ and tangent to $ BC$ at $ D$. Let $ AD$ meets the circumcircle of $ ABC$ agin at $ Q$. Let $ O$ be the circumcenter of triangle $ ABC$. If the line $ AO$ bisects $ \angle DAC$, prove that the circle centered at $ Q$ passing through $ B$, circle $ P$, and the perpendicular line of $ AD$ from $ B$, are all concurrent.
2003 France Team Selection Test, 3
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2022 Brazil Team Selection Test, 2
Let $ABC$ be a triangle with orthocenter $H$, $\Gamma$ its circumcircle, and $A' \neq A$, $B' \neq B$, $C' \neq C$ points on $\Gamma$. Define $l_a$ as the line that passes through the projections of $A'$ over $AB$ and $AC$. Define $l_b$ and $l_c$ similarly. Let $O$ be the circumcenter of the triangle determined by $l_a$, $l_b$ and $l_c$ and $H'$ the orthocenter of $A'B'C'$. Show that $O$ is midpoint of $HH'$.
2016 Kyrgyzstan National Olympiad, 6
Given three pairwise tangent equal circles $\Omega_i (i=1,2,3)$ with radius $r$.
The circle $\Gamma $ touches the three circles internally (circumscribed about 3 circles).The three equal circles $\omega_i (i=1,2,3)$ with radius $x$ touches $\Omega_i$ and $\Omega_{i+1}$ externally ($\Omega_4= \Omega_1$) and touches $\Gamma$ internally.Find $x$ in terms of $r$
2010 Contests, 3
Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.
2020 Balkan MO Shortlist, G3
Let $ABC$ be a triangle. On the sides $BC$, $CA$, $AB$ of the triangle, construct outwardly three squares with centres $O_a$, $O_b$, $O_c$ respectively. Let $\omega$ be the circumcircle of $\vartriangle O_aO_bO_c$. Given that $A$ lies on $\omega$, prove that the centre of $\omega$ lies on the perimeter of $\vartriangle ABC$.
[i]Sam Bealing, United Kingdom[/i]
2022 Pan-African, 1
Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$, and $AB$ its shortest side. Let $H$ be the orthocenter of $ABC$. Let $\Gamma$ be the circle with center $B$ and radius $BA$. Let $D$ be the second point where the line $CA$ meets $\Gamma$. Let $E$ be the second point where $\Gamma$ meets the circumcircle of the triangle $BCD$. Let $F$ be the intersection point of the lines $DE$ and $BH$.
Prove that the line $BD$ is tangent to the circumcircle of the triangle $DFH$.
2003 France Team Selection Test, 3
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2010 India National Olympiad, 5
Let $ ABC$ be an acute-angled triangle with altitude $ AK$. Let $ H$ be its ortho-centre and $ O$ be its circum-centre. Suppose $ KOH$ is an acute-angled triangle and $ P$ its circum-centre. Let $ Q$ be the reflection of $ P$ in the line $ HO$. Show that $ Q$ lies on the line joining the mid-points of $ AB$ and $ AC$.
2008 Costa Rica - Final Round, 2
Let $ ABC$ be a triangle and let $ P$ be a point on the angle bisector $ AD$, with $ D$ on $ BC$. Let $ E$, $ F$ and $ G$ be the intersections of $ AP$, $ BP$ and $ CP$ with the circumcircle of the triangle, respectively. Let $ H$ be the intersection of $ EF$ and $ AC$, and let $ I$ be the intersection of $ EG$ and $ AB$. Determine the geometric place of the intersection of $ BH$ and $ CI$ when $ P$ varies.
1974 IMO Longlists, 15
Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$.
[hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]
2021 Iran MO (2nd Round), 3
Circle $\omega$ is inscribed in quadrilateral $ABCD$ and is tangent to segments $BC, AD$ at $E,F$ , respectively.$DE$ intersects $\omega$ for the second time at $X$. if the circumcircle of triangle $DFX$ is tangent to lines $AB$ and $CD$ , prove that quadrilateral $AFXC$ is cyclic.
2017 Polish MO Finals, 1
Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.